Calculation of Total, Systematic, and Unsystematic Risk

Quantifying Investment Risk

1. Total Risk:

• Definition: Represents the overall variability or uncertainty of an investment's returns. It encompasses both systematic and unsystematic risk.

• Measurement: Standard Deviation (σ) of returns.

  • Standard deviation measures the dispersion of returns around the average return. A higher standard deviation indicates greater volatility and, therefore, higher total risk.
  • σ = square root of ( Σ ((R_i - R)^2) / (n - 1) )
    • R_i is each item being measured.
    • R is the mean.
    • n is the number of items being measured.
    • Σ means to perform the summation.

• Utility: Provides a comprehensive measure of the investment's overall riskiness.

2. Systematic Risk:

• Definition: The portion of an investment's total risk that is attributable to factors affecting the entire market. This risk cannot be diversified away.

• Measurement: Beta (β)

  • Beta measures the sensitivity of an investment's returns to movements in the overall market (or a specific benchmark index).
  • A beta of 1 indicates that the investment's price tends to move in the same direction and magnitude as the market.
  • A beta greater than 1 indicates that the investment is more volatile than the market.
  • A beta less than 1 indicates that the investment is less volatile than the market.
  • A negative beta would correlate that the instrument usually moves in the opposite direction as that market.

3. Unsystematic Risk:

• Definition: The portion of an investment's total risk that is attributable to factors specific to the company or industry. This risk can be reduced through diversification.

• Formula (as provided):

  • Unsystematic Risk = Total Risk - Systematic Risk

• Important Note: This is an oversimplification. You can't directly subtract Beta from Standard Deviation.

The correct formula for calculating Unsystematic Risk (expressed as variance) is:

• Unsystematic Risk is Total Risk minus Systematic Risk: σ²(Unsystematic Risk) = σ²(Total Risk) - β² * σ²(Market)

• From here, you square root the Unsystematic Risk to arrive at the Unsystematic Standard Deviation.

  • The book did note that the relation between risk and return could be expressed more clearly by combining the three. The given formula only looks at risk, as separate from return.

Using the Formulas Correctly:

1. Calculate Total Risk (σ): Calculate the standard deviation of the investment's returns using historical data.
2. Calculate Systematic Risk (β): Determine the investment's beta, which measures its sensitivity to market movements.
3. Calculate Market Risk (σm): Find the square of the standard deviation for the market’s return.
4. Then, calculate Unsystematic risk using the adjusted formula: Calculate the variance and root it: sqrt[ sqrt( σ²) - (β² + σ(market)) ]

In Summary:

The breakdown into Total, Systematic, and Unsystematic risk provides a powerful framework for understanding and managing investment risk. While total risk provides a broad overview, understanding systematic and unsystematic components allows investors to diversify effectively and to focus on managing the specific risks that are most relevant to their portfolios. Again, a great portfolio manager will understand this and be able to advise as to these risks to their investors.